Shrinkage Estimators of BLUE for Time Series Regression Models

The least squares estimator (LSE) seems a natural estimator of linear regression models. Whereas, if the dimension of the vector of regression coefficients is greater than 1 and the residuals are dependent, the best linear unbiased estimator (BLUE), which includes the information of the covariance matrix \(\boldsymbol{\Gamma }\) of residual process has a better performance than LSE in the sense of mean square error. As we know the unbiased estimators are generally inadmissible, Senda and Taniguchi (2006) introduced a James-Stein type shrinkage estimator for the regression coefficients based on LSE, where the residual process is a Gaussian stationary process, and provides sufficient conditions such that the James-Stein type shrinkage estimator improves LSE. In this chapter, we propose a shrinkage estimator based on BLUE. Sufficient conditions for this shrinkage estimator to improve BLUE are also given. Furthermore, since \(\boldsymbol{\Gamma }\) is infeasible, assuming that \(\boldsymbol{\Gamma }\) has a form of \(\boldsymbol{\Gamma }=\boldsymbol{\Gamma }(\boldsymbol{\theta })\) , we introduce a feasible version of that shrinkage estimator with replacing \(\boldsymbol{\Gamma }(\boldsymbol{\theta })\) by \(\boldsymbol{\Gamma }(\hat{\boldsymbol{\theta }})\) which is introduced in Toyooka (1986). Additionally, we give the sufficient conditions where the feasible version improves BLUE. Besides, the results of a numerical study confirm our approach. This chapter is mainly based on Xue et al. (2024).